Implications

We have the following indirect implication of form equivalence classes:

384 \(\Rightarrow\) 406
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
384 \(\Rightarrow\) 14 "Maximal filters, continuity and choice principles", Herrlich, H. 1997, Quaestiones Math.
14 \(\Rightarrow\) 385 clear
385 \(\Rightarrow\) 406 The axiom of choice and two particular forms of Tychonoff theorem, Alas, O. T. 1969, Portugal. Math.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
384:

Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter.

14:

BPI: Every Boolean algebra has a prime ideal.

385:

Countable Ultrafilter Theorem:  Every proper filter with a countable base over a set \(S\) (in \({\cal P}(S)\)) can be extended to an ultrafilter.

406:

The product of compact Hausdorf spaces is countably compact. Alas [1994].

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